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p-adic Strings: Thermal Duality & the Cosmological Constant
Tirthabir Biswas
Loyola University, New Orleans
PRL 104, 021601 (2010) JHEP 1010:048, (2010)PRD 82:085028, (2010) with J. Kapusta and J .A. R. Cembranos.
1
2
2
2 1
1exp
2
1 pD
p
Ds
pMxd
g
mS
'2
1
ln
2
1
11 22
22
22
s
s
op
mp
mM
p
p
gg
The N-point tree amplitudes of the open string can begenerated from a non-local Lagrangian of a single scalarfield.
Volovich, Brekke, Freund, Olson, Witten, Frampton, Okada, late 80’s
open string coupling prime number string tension
p-adic Action
Generalizations Nonlocal Infinite derivative Actions
p-adic theories Strings on Random lattice (Douglas&Shenker,Gross&Migdal, 1990)
Regge trajectories (TB, Siegel, Grisaru 2004)
String Field Theories
Related cousins appear in Noncommutative Field theory Theory of unparticles
)( VFxdS D O
)exp()()( 2 OOO mF
What can we gain? Insights into string theory Hagedorn physics Brane Physics
Applications to Cosmology Novel kinetic energy dominated non-slow-roll inflationary
mechanisms (TB, Barnaby, Cline, 2006), large nongaussianities (Barnaby,Cline, 2007)
Dark Energy (Arefeva et.al.)
Thermal Cosmology in the Early Universe
Applications to Particle Physics [Moffat et.al.]
Interesting Properties (?) Usually higher derivative theories are plagued by ghosts:
But padic type theories have no extra states! Initial value problem may be well defined: Studied by Russian mathematicians, 2 degrees of freedom for every
pole, rigorously established for free theories. [Barnaby, Kamran] Diffusion equation formulation in one higher dimension suggest
finite number of IC’s. [Calcagni, Nardelli...]
No perturbative states, free theory is trivial, quantum contributions only arise when interactions are present.
Field Equations can be recast as integral equations and hence numerical progress/tests can be made.
• Rescale the fields to put the action in the form
1
2
2223 /
exp2
1 p
M
dxddS
1
21
1
p
s
p
m
g
p
with dimensionful coupling constant
and non-local propagator nT
MkkD
n
nn
2
/exp),( 2220
Thermal Field Theory
2
03
3
2 ),()2(
)3(ln
nn kD
kdTVZ
M
TN
N
MkD
kd
nn
N
2
2),(
)2(
3
03
3
p=3
xxex
n
xn 22
)(
2-loop & Thermal Duality
Compute Feynman diagrams
Partition Function
Analogous to t-duality
This was conjectured using “real string theory” arguments (momentum modes and winding modes)
It was also conjectured that non-perturbative corrections will violate the duality In the p-adic case, this happens at higher loops
2 )( 0
20
11
MT
T
TZTZ
242
1
22
43
M
TN
M
TMP
xxex
n
xn 22
)(
22 /21)( xex
x
2
21)( xex
Thermodynamics
• Low Temperature:
Behaves as pressureless dust (Deo et. al.,Vafa,Tseythlin, Brandenberger)
• High Temperature (Atick & Witten):
Behaves as stiff fluid
02
exp1
2exp
2
2
22
2
T
M
Tand
T
MP
12 TP
Vacuum Energy Vacuum energy appears only at 2-loops It is -ve Ghost appears due to self-energy
Adding counter-term makes the vacuum energy positive and hierarchically suppressed
122
2
1
4!!)1( with term-counter theAdd
pM
pp
.0at on contributienergy -self thecancel to T
2/)1(
03
3
1 ),()2(
!!)1(
p
nn kD
kdTpp
04
!!)112
(2
1vac
pM
pp
Planck Mass & Cosmological Constant
21926
862 GeV 1022.1
21
o
s
NP g
mV
GM
3
)(
p
P
s
M
mpc hierarchical suppression
known dimensionless function of p
volume of extra-dimensional compactified space
p=7
PeV 385m7p
TeV 1820m5p
MeV 550m3p
then )meV 3.2( If
s
s
s4
vac
Necklace diagrams
and sunset diagrams
Low T High T (Atick & Witten) OK
.2o
s
g
mT
Higher Loops
)1(2
)1(2 ~ln~ln
l
s
sl
lsl m
TgZgZ
Solitons at Finite temperature
maxmin fff
01 & 1
:1 Around
1 & 1 :0 Around
1 & 10 :1 Around
maxmin
maxmin
maxmin
ff
ff
ff
ppp ,7,3
p
M
2
222 /exp Classical Equation:
0at Gaussian T
cosine
2ln
for 1pM
TTf c
even solution p=3
95.0,85.0,5.0,3.0,1.0,01.0/ cTT
odd solution p=3
50.1,0.1,5.0,1.0,01.0/ cTT
0at Gaussians T
T as amplitude
increasing with sine
ESeVZ solitonln )()(
2TI
g
paS
oE
Even solitons are important for high temperatures
22 /in lexponentia MT
TTc /
Future Applications
Thermal fluctuations may dominate the early phase of inflation…any signatures (?)
Thermal solitons suggest existence of branes in extra compact directions
Exponential cut-off acts like a regularization parameter – can it be made physical?
What about bound states?What about Gravity? (TB,Mazumdar,Siegel’05)
In mathematics, and chiefly number theory, the p-adic number systemfor any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational numbersystem to the real and complex number systems. The extension isachieved by an alternative interpretation of the concept of absolute value.
Wikipedia
String Theories over p-adic Fields*
• Quantum fields valued in the field of complex numbers• Space-time coordinates valued in the field of real numbers• World-sheet coordinates valued in the field of p-adic numbers
*Freund & Olson (1987), Freund & Witten (1987)
2
12222
4!!
2/)1(12
1
p
M
T
M
T
M
T
M
TMpP
pp
04
!!)112
(2
1vac
pM
pp
04
!! 2/)1(
4/)1(32
1
p
p
TM
pP
vac2
2122
2
11 2
exp4
!!1
T
MMppP
p
vacuum energy:
low T:
high T:
no particle degreesof freedom
Solitons at Finite Temperature
p
M
2
222 /exp :motion ofequation classical
pxMppxf
xfxffxx
ps
nssn
4/)1(exp)(
)()()(),...,(
22)1(2/1
11
. period with periodic bemust )(function The f
.1,0 are solutions Trivial
Soliton solutions in Euclidean space at zero temperaturewere found by Brekke, Freund, Olson & Witten (1988).
Key Results
• There are no particle degrees of freedom so there is no one-loop contribution to the partition function
• The lowest order contribution arises from interactions• A counter-term must be added to avoid the appearance
of a ghost in a loop expansion which has the consequence that …
• The vacuum energy is positive and hierarchically suppressed
• Perturbation theory breaks down at a temperature of order
• Soliton solutions exist at all temperatures and become important when
2/~ osc gmT
2/ os gmT